What means the velocity constant of the integration in PI controller formulas?

Question

I have an equation of a PI controller. What means here the velocity constant "R" exactly?

$$m(t)=K(e(t)+R{\int}e(t)dt)$$

$$R=1sec^{-1}$$

Answer 1

Quick Answer

The $R$ in the PI controller equation would best be described as a "weighting factor" that determines the influence of the integral term relative to the proportional term. The larger $R$ gets, the more the integrated error affects the output of the controller compared to the instantaneous error.

More explanation

Often PI controllers are expressed in the following form:

$u(t) = K e(t) + K_I \int e(t)dt$

where $K$ is called the proportional gain, and $K_I$ is called the integral gain. You tune the controller by changing $K$ and $K_I$ until the system behaves the way you want.

The equation given in the question expresses $K_I$ as a multiple of $K$, i.e. $K_I = RK$. Therefore if you substitute it back into the controller equation:

$u(t) = K e(t) + RK \int e(t)dt = K (e(t) + R \int e(t)dt)$

Now, instead of tuning $K$ and $K_I$ separately, you can look at the process as tuning the proportional gain $K$, then deciding how much influence the integral term has by adjusting $R$. This way of expressing the gains might be useful depending on how you approach the problem of tuning the controller. Really it's just mathematical semantics, you can use whichever form makes more sense to you.

Aside

$R$ has units of $\text{s}^{-1}$ (inverse seconds) in order to make the equation have consistent units. For example if $e(t)$ is a distance, then $\int e(t) dt$ has units of distance * time. Therefore, $R$ must have units of 1/time in order to cancel the time in the integral term.